Lattice $\phi^4$ Field Theory on Riemann Manifolds: Numerical Tests for the 2-d Ising CFT on $\mathbb{S}^2$

TL;DR

This paper develops a numerical method to realize lattice $^4$ quantum field theory on Riemann manifolds, specifically testing it on the 2D sphere and comparing results with exact Ising CFT solutions.

Contribution

It introduces a novel approach combining Regge Calculus and finite element methods for lattice field theories on curved manifolds, enabling numerical tests against known CFT results.

Findings

Numerical results agree with exact Ising CFT solutions

Binder cumulants up to 12th order match theoretical predictions

Correlation functions are consistent with continuum theory

Abstract

We present a method for defining a lattice realization of the $\phi^4$ quantum field theory on a simplicial complex in order to enable numerical computation on a general Riemann manifold. The procedure begins with adopting methods from traditional Regge Calculus (RC) and finite element methods (FEM) plus the addition of ultraviolet counter terms required to reach the renormalized field theory in the continuum limit. The construction is tested numerically for the two-dimensional $\phi^4$ scalar field theory on the Riemann two-sphere, $\mathbb{S}^2$, in comparison with the exact solutions to the two-dimensional Ising conformal field theory (CFT). Numerical results for the Binder cumulants (up to 12th order) and the two- and four-point correlation functions are in agreement with the exact $c = 1/2$ CFT solutions.